| 1. |
- Adlerborn, Björn, et al.
(författare)
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A parallel QZ algorithm for distributed memory HPC systems
- 2014
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Ingår i: SIAM Journal on Scientific Computing. - : SIAM publications. - 1064-8275 .- 1095-7197. ; 36:5, s. C480-C503
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Tidskriftsartikel (refereegranskat)abstract
- Appearing frequently in applications, generalized eigenvalue problems represent one of the core problems in numerical linear algebra. The QZ algorithm of Moler and Stewart is the most widely used algorithm for addressing such problems. Despite its importance, little attention has been paid to the parallelization of the QZ algorithm. The purpose of this work is to fill this gap. We propose a parallelization of the QZ algorithm that incorporates all modern ingredients of dense eigensolvers, such as multishift and aggressive early deflation techniques. To deal with (possibly many) infinite eigenvalues, a new parallel deflation strategy is developed. Numerical experiments for several random and application examples demonstrate the effectiveness of our algorithm on two different distributed memory HPC systems.
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| 2. |
- Adlerborn, Björn, et al.
(författare)
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Towards Highly Parallel and Compute-Bound Computation of Eigenvectors of Matrices in Schur Form
- 2017
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we discuss the problem of computing eigenvectors for matrices in Schur form using parallel computing. We develop a new parallel algorithm and report on the performance of our MPI based implementation. We have also implemented a new parallel algorithm for scaling during the backsubstitution phase. We have increased the arithmetic intensity by interleaving the compution of several eigenvectors and by merging the backward substitution and the back-transformation of the eigenvector computation.
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| 3. |
- Dackland, Krister, et al.
(författare)
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A ring-oriented approach for block matrix factorizations on shared and distributed memory architectures
- 1993
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Ingår i: Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing. - Norfolk : SIAM Publications. - 0898713153 ; , s. 330-338
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Konferensbidrag (refereegranskat)abstract
- A block (column) wrap-mapping approach for design of parallel block matrix factorization algorithms that are (trans)portable over and between shared memory multiprocessors (SMM) and distributed memory multicomputers (DMM) is presented. By reorganizing the matrix on the SMM architecture, the same ring-oriented algorithms can be used on both SMM and DMM systems with all machine dependencies comprised to a small set of communication routines. The algorithms are described on high level with focus on portability and scalability aspects. Implementation aspects of the LU , Cholesky, and QR factorizations and machine specific communication routines for some SMM and DMM systems are discussed. Timing results show that our portable algorithms have similar performance as machine specific implementations. 1 Introduction With the introduction of advanced parallel computer architectures a demand for efficient and portable algorithms has emerged. Several attempts to design algorithms and implementat.
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| 6. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Codimension computations of congruence orbits of matrices, symmetric and skew-symmetric matrix pencils using Matlab
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Matlab functions to work with the canonical structures for congru-ence and *congruence of matrices, and for congruence of symmetricand skew-symmetric matrix pencils are presented. A user can providethe canonical structure objects or create (random) matrix examplesetups with a desired canonical information, and compute the codi-mensions of the corresponding orbits: if the structural information(the canonical form) of a matrix or a matrix pencil is known it isused for the codimension computations, otherwise they are computednumerically. Some auxiliary functions are provided too. All thesefunctions extend the Matrix Canonical Structure Toolbox.
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| 7. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Geometry of Matrix Polynomial Spaces
- 2020
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Ingår i: Foundations of Computational Mathematics. - : Springer-Verlag New York. - 1615-3375 .- 1615-3383. ; 20:3, s. 423-450
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Tidskriftsartikel (refereegranskat)abstract
- We study how small perturbations of general matrix polynomials may change their elementary divisors and minimal indices by constructing the closure hierarchy (stratification) graphs of matrix polynomials’ orbits and bundles. To solve this problem, we construct the stratification graphs for the first companion Fiedler linearization of matrix polynomials. Recall that the first companion Fiedler linearization as well as all the Fiedler linearizations is matrix pencils with particular block structures. Moreover, we show that the stratification graphs do not depend on the choice of Fiedler linearization which means that all the spaces of the matrix polynomial Fiedler linearizations have the same geometry (topology). This geometry coincides with the geometry of the space of matrix polynomials. The novel results are illustrated by examples using the software tool StratiGraph extended with associated new functionality.
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| 8. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Symmetric matrix pencils : codimension counts and the solution of a pair of matrix equations
- 2014
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Ingår i: The Electronic Journal of Linear Algebra. - : University of Wyoming Libraries. - 1537-9582 .- 1081-3810. ; 27, s. 1-18
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Tidskriftsartikel (refereegranskat)abstract
- The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.
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| 9. |
- Edelman, Alan, et al.
(författare)
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A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part I: Versal Deformations
- 1997
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Ingår i: SIAM Journal on Matrix Analysis and Applications. ; 18:3, s. 653-692
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Tidskriftsartikel (refereegranskat)abstract
- We derive versal deformations of the Kronecker canonical form by deriving the tangent space and orthogonal bases for the normal space to the orbits of strictly equivalent matrix pencils. These deformations reveal the local perturbation theory of matrix pencils related to the Kronecker canonical form. We also obtain a new singular value bound for the distance to the orbits of less generic pencils. The concepts, results, and their derivations are mainly expressed in the language of numerical linear algebra. We conclude with experiments and applications.
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| 10. |
- Edelman, Alan, et al.
(författare)
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A Geometric Approach to Perturbation Theory of Matrices and Matrix Pencils. Part II: A Stratification-Enhanced Staircase Algorithm
- 1999
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Ingår i: SIAM Journal on Matrix Analysis and Applications. ; 20:3, s. 667-699
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Tidskriftsartikel (refereegranskat)abstract
- Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but it may also be read independently.
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